Printing playing cards in Budapest

Try copyguru or copy general, they have web sites with contact info.  If you need it in HUGE quantities, a real printery might be cheaper, google for "budapest nyomda".

Going to a copy store probably would be the best place to start if you are only printing out a few decks of playing cards.
It would cost a small fortune to get real playing cards made up, packaged etc. You would want to make several thousand decks I would think if you wanted high quality cards made up.
I worked for 3 days as a rep. for Angel Playing Cards at the Las Vegas convention center when I was still in dealers school.
It is a family co. from Japan, they were trying to sell the very beautiful and high quality playing cards to casinos.
Of course I don't know what happened behind closed doors with their business deals but their cards were so nice, smooth and easy to handle that their prices were probably a bit too high for most casinos to purchase. Casinos change out their playing cards at least 3 times a day on every game, sometimes more if drinks spill on them or players bend the cards, all depends on the game, one reason casinos do not always let players handle the cards as in open faced blackjack.
Most casinos do not buy nice opaque cards, they go for the cheapest they can buy in bulk, ACE is usually the brand they use in Vegas.
If you plan on mass production of your game then it will take a bit more research to find a manufacturer who can print out high quality cards for a decent price.
Print shop is a good start, they may even know of a place that can put a glossy finish on your cards if they can't do it themselves.
The cards produced by Angel Cards were actually too pretty to use except for a private game, some were plastic cards, opaque finish on some too. Must of been very pricey to produce them.
Lucky me, they gave me a dozen or so decks as a gift for working for them.
So best bet is trying a copy shop, if you want plastic opaque cards then you will have to look into someplace that works with plastic materials, no idea where to start with that one.
Getting printed out in paper first sounds like a better plan , allot cheaper then plastic prints. See how popular your new game is before mass production.
My job as a card rep. was really nothing to do with manufacturing , more about selling.
They called all the schools in Vegas and asked for a student dealer to show how the cards looked and to set up a game table and let buyers experience the way the cards held up on their own with use. It was fun and a bit of a ego boost that they chose me out of all the other students in class. Guess it is always best to "put your best foot forward" in public at all times and to always look the part.The pay was fantastic for 3 short days, had to get listed into a "talent agency" to be hired.
Vegas, a very strange city.
Just saying, not easy or cheap to get your product out there to the public, takes allot  of money to start.